Confidence Interval for a Mean

A confidence interval for a mean is a range of values that is likely to contain the true population mean with a certain level of confidence. It provides both an estimate of the mean and a measure of the uncertainty associated with that estimate.

The general formula for a confidence interval is:

Where:

• $\bar x$ is the sample mean
• The critical value depends on the confidence level and whether you're using a z-distribution or t-distribution
• The standard error is $\frac{s}{\sqrt{n}}$ for unknown population standard deviation, or $\frac{\sigma}{\sqrt{n}}$ for known population standard deviation

If you were to repeat the sampling process many times and calculate a 95% confidence interval for each sample, about 95% of these intervals would contain the true population mean. It does not mean there's a 95% probability that the population mean falls within a single calculated interval.

• The sample is random and representative of the population
• The data is normally distributed or the sample size is large enough ($n \gt 30$)
• For small samples, the population should be at least 20 times larger than the sample